Martin's Blog

Endomorphisms of simple abelian varieties

Posted by Martin Orr on Thursday, 03 January 2013 at 15:51

Today I will discuss the classification of endomorphism algebras of simple abelian varieties. The endomorphism algebra of a non-simple abelian variety can easily be computed from the endomorphism algebras of its simple factors. For a simple abelian variety, its endomorphism algebra is a division algebra of finite dimension over \mathbb{Q}. (A division algebra is a not-necessarily-commutative algebra in which every non-zero element is invertible.) As discussed last time, the endomorphism algebra also has a positive involution, the Rosati involution. There may be many Rosati involutions, coming from different polarisations of the abelian variety, but all we care about today is the existence of a positive involution. Division algebras with positive involutions were classified by Albert in the 1930s.

The table lists the four types of endomorphism algebras of simple abelian varieties. Here, D = \operatorname{End} A \otimes_\mathbb{Z} \mathbb{Q} and F is the centre of D. The index of D is \sqrt{\dim_F D} where F is the centre of D (\dim_F D is always a square because D \otimes_F \bar{F} is a matrix algebra over \bar{F}). We write d for the index of D and e = [F \colon \mathbb{Q}]. The "involution type" column is explained below.

In the dimension conditions, g is the dimension of our abelian variety. These conditions are valid only for abelian varieties defined over fields of characteristic zero. All the rest of the table applies to abelian varieties defined over fields of any characteristic. There are weaker dimension conditions in positive characteristic (they are listed in Mumford, which I do not have to hand right now).

Albert type Centre Involution type Index D \otimes_\mathbb{Q} \mathbb{R} Dimension conditions
I totally real orthogonal 1 \mathbb{R}^e e|g
II totally real orthogonal 2 \mathrm{M}_2(\mathbb{R})^e 2e|g
III totally real symplectic 2 \mathbb{H}^e 2e|g
IV CM unitary d \mathrm{M}_d(\mathbb{C})^{e/2} \frac{1}{2} e d^2 | g

Division algebras with involution

Division algebras with involution are divided into two cases depending on the behaviour of the involution on the centre. Let (D, \dag) be a division algebra with involution and F its centre. Then F is a field, and \dag restricts to an automorphism of F. Let F_0 be the subfield of F fixed by \dag. Because \dag\dag is the identity, either F = F_0 or [F:F_0] = 2. These two cases are called involutions of the first kind and second kind respectively.

We can fully classify division algebras with involutions of the first kind when F is a number field, giving types I, II and III in the Albert classification. Involutions of the second kind might look simpler because they give only Albert type IV, but really there are just too many of them to classify further.

Involutions of the first kind

If \dag is an involution of the first kind, then we can extend it \bar{F}-linearly to an involution of D \otimes_F \bar{F}.

By the Artin-Wedderburn theorem, D \otimes_F \bar{F} is isomorphic to \mathrm{M}_d(\bar{F}) for some n. As we saw last week, the antiautomophism \dag of \mathrm{M}_d(\bar{F}) corresponds to a bilinear form \psi on \bar{F}^d (defined up to multiplication by scalars). Since \dag is an involution, the bilinear form is either symmetric or alternating. Hence we can divide involutions of the first kind into two classes: involutions of orthogonal type, which correspond to a symmetric form on \bar{F}^d, and involutions of symplectic type, which correspond to an alternating form \bar{F}^d.

The Brauer group and involutions of the first kind

We can further constrain algebras with an involution of the first kind by looking at the Brauer group of F. This is the group whose elements are isomorphism classes of division algebras with centre F and whose multiplication is defined as follows: if D_1, D_2 are division algebras with centre F then by Artin-Wedderburn, D_1 \otimes_F D_2 \cong \mathrm{M}_n(D_3) for some division algebra D_3 with centre F. We define [D_1].[D_2] = [D_3] in the Brauer group.

The inverse of a division algebra in the Brauer group is the opposite algebra, that is the algebra with the same elements but with the order of multiplication reversed. An involution of an algebra D fixing F is an isomorphism of F-algebras D \to D^\mathrm{op}, so any algebra with involution of the first kind must satisfy [D] = [D]^{-1} in \operatorname{Br} F.

Hence either [D] = 1 in the Brauer group or [D] is an element of order 2. If [D] = 1 then D = F (and the involution must be of orthogonal type, because there is no non-degenerate alternating form on a space of dimension 1). This is Type I in the Albert classification.

So far our classification has been valid for division algebras over any field. Now we use the fact that F is a number field. Class field theory implies that the index of a division algebra over a number field is equal to its order in the Brauer group (a corollary of the Albert-Brauer-Hasse-Noether theorem). So if [D] has order 2 then D is a quaternion algebra over F. (Over a general field F, [D] having order 2 only implies that the index is a power of 2.) According to whether the involution is of orthogonal type or symplectic type, we get Types II and III in the Albert classification.

Conversely, given any quaternion algebra, there exists a unique involution of symplectic type and a conjugacy class of involutions of orthogonal type.

Involutions of the second kind

Now consider an involution of the second kind. Let \sigma be its restriction to F. Then \sigma does not always extend to an automorphism of \bar{F} of order 2, so \dag need not extend to D \otimes_F \bar{F}. However if \sigma can be extended to an automorphism of \bar{F} of order 2, then \dag can be extended to D \otimes_F \bar{F}. Then \dag will be the adjoint involution of \mathrm{M}_d(\bar{F}) corresponding to some Hermitian form on \bar{F}^d, similarly to how involutions of the first kind are related to bilinear forms. Hence the are sometimes called involutions of unitary type. (If the involution is positive, then its restriction to F is complex conjugation, as we will see below, so it does always extend to \bar{F}.)

A division algebra with an involution of the second kind satisfies D^\mathrm{op} \cong D^\sigma, where D^\sigma is the Galois conjugate of D. This does not restrict the order of [D] in the Brauer group, so the index of D can be unbounded. Also the condition D^\mathrm{op} \cong D^\sigma is not sufficient condition for the existence of an involution of the second kind.

Positive involutions

So far we have only classified division algebras with involutions, without using the positivity of the involution. The positivity of the involution is needed to get the columns on the centre and on D \otimes_\mathbb{Q} \mathbb{R} in the table above. Arguments using the weak approximation lemma show that any field on which the identity is a positive involution must be totally real, while any field with a non-trivial positive involution is a CM field (this also establishes that a non-trivial positive involution of a field must be the restriction of complex conjugation).

In order to determine D \otimes_\mathbb{Q} \mathbb{R} for types II and III, we note first that it must be a product of copies of \mathrm{M}_2(\mathbb{R}) and the Hamilton quaternions \mathbb{H}. Then verify that \mathrm{M}_2(\mathbb{R}) has only positive involutions of orthogonal type, while \mathbb{H} has only a positive involution of symplectic type.

The dimension bounds

Most of the dimension bounds come from the fact that \operatorname{End} A \otimes \mathbb{Q} acts faithfully on H_1(A, \mathbb{Z}) \otimes \mathbb{Q}. (This works only over fields of characteristic zero because H_1(A, \mathbb{Z}) is only defined in characteristic zero. In positive characteristic we would have to use the \ell-adic Tate modules instead, and then D \otimes \mathbb{Q}_\ell need not be a division algebra.) Hence H_1(A, \mathbb{Z}) \otimes \mathbb{Q} is a "D-vector space" and like a vector space over a field, it must be isomorphic to D^n for some integer n. Hence \dim_\mathbb{Q} D = ed^2 divides \dim_\mathbb{Q} H_1(A, \mathbb{Z}) \otimes \mathbb{Q} = 2g. This gives the bounds for types II, III and IV.

In order to get e|g instead of e|2g for type I, we need to use the fact that the action of \operatorname{End} A on H_1(A, \mathbb{Z}) \otimes \mathbb{R} commutes with the complex structure on the latter. First, let F = \operatorname{End} A \otimes \mathbb{Q} and V = H_1(A, \mathbb{Z}) \otimes \mathbb{Q}. We know that F \otimes_\mathbb{Q} \mathbb{R} \cong \mathbb{R}^e so V_\mathbb{R} splits into e subrepresentations V_\sigma of F \otimes \mathbb{R}, one for embedding \sigma \colon F \hookrightarrow \mathbb{R}. The Galois group \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) permutes the V_\sigma transitively, so they all have the same dimension.

Since the action of F on V_\mathbb{R} commutes with the complex structure, each subrepresentation V_\sigma is a sub-complex structure and so has even dimension. So 2g = \dim V = e.\dim V_\sigma and \dim V_\sigma is even. This implies that e divides g.

Tags abelian-varieties, alg-geom, maths

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